How to use normal maps with nodes in Blender Internal?
In order to use normal map in BI in particular you should have appropriate image - normal map. 

Black&amp;White image won't work for this purpose, it will work as a bump map (which is one used on Cube A in your example). Normal map uses RGB values to convert into orientation of given point on the surface; black&amp;white image doesn't provide that information.
Normal map is an image in blueish hues which tells a given 'face' which orientation should it have with respect to its position within the overall structure of the object, and here's an example of one (with respect to this case):



There are different ways to create normal map based on this image, see How do I make a normal/displacement map when all I have is a photo?.
You can create it in Blender using the same way as in normal maps baking from sculpting with the only difference that image will be used instead. This may be not the best way, although it works for fast and dirty results. 

Once you have that image use it as a texture for material of the object:



For more details about using normal maps in Blender Internal and Cycles see:


http://thezedlab.com/quick-reference-using-normal-maps-in-bi-and-cycles/
http://www.katsbits.com/tutorials/blender/bake-normal-maps.php